Question

Question 1. We wish to characterise the risk of default (going bankrupt) of individual companies Data are available on the survival times in years), that is, time until default. We model the survival times as realisations from an independent and identically distributed (TID) exponen- tial random variable with mean 1/0 and density f(310) Be for 0 > 0,2 > 0. (i) Calculate the maximum likelihood estimate (MLE) for 8 from the data 11,-. In (1) Calculate the expected information for 8, and hence the asymptotic variance of the MLE for 8. (lil) Let be the probability of surviving longer than 3 years. Write as a function of and give the MLE for o. Compute a 95% confidence interval for (iv) It turns out that some of the companies were still alive at the time of recording so that those recorded times were not actual survival times but the last observations available. Briefly explain how this would affect the inference on the parameter 8.

Answer 1

Solution f(x/6)= 0 2 0%, 0%0,n30 I :) MLE for a O>o, uso falo)o é o x, 070,470 Uxle) on ēossi log L(1/0)= nlogo - oexi odlog u ulo-0 -Sri 30 -0 n rexi =Exi Erci ! Ô = I z > llit is the AMLE of ois Milie of on 11) Information for a Ilo)= E(-ology I(o)=E(-3 lagf)

From ② , - pole = Ô Eri Eni ao for any en que t e 1) Q = Probability of surinting surviving longer than 3 years $=P(x+3) = F (3) &= F(x)= 1- en - F(3) = 1-630 asl. Confidence interval for & 1280=4 -20 21-01 -30=log (1-21)

ôc = I log (1-4) H 2 = 1-da -30 = logos êu = 1 log da 95% confidence Interval for p is (togenas), og da ( 7 logoros, Flag ons) iv) Exponential distribution has loss of memory property . P(x?tts| xət) = f(x2) Hence the survival times mes of of the ope the previous observations are not dependent on latest Oles ervations. J